A356844 Numbers k such that the k-th composition in standard order contains at least one 1. Numbers that are odd or whose binary expansion contains at least two adjacent 1's.

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87

Offset

1,2

Comments

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Links

Table of n, a(n) for n=1..67.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Formula

Union of A005408 and A004780.

Example

The terms, binary expansions, and standard compositions:
   1:       1  (1)
   3:      11  (1,1)
   5:     101  (2,1)
   6:     110  (1,2)
   7:     111  (1,1,1)
   9:    1001  (3,1)
  11:    1011  (2,1,1)
  12:    1100  (1,3)
  13:    1101  (1,2,1)
  14:    1110  (1,1,2)
  15:    1111  (1,1,1,1)
  17:   10001  (4,1)
  19:   10011  (3,1,1)
  21:   10101  (2,2,1)
  22:   10110  (2,1,2)
  23:   10111  (2,1,1,1)
  24:   11000  (1,4)
  25:   11001  (1,3,1)
  26:   11010  (1,2,2)
  27:   11011  (1,2,1,1)
  28:   11100  (1,1,3)
  29:   11101  (1,1,2,1)
  30:   11110  (1,1,1,2)
  31:   11111  (1,1,1,1,1)

Mathematica

Select[Range[0, 100], OddQ[#]||MatchQ[IntegerDigits[#, 2], {___, 1, 1, ___}]&]

Crossrefs

See link for sequences related to standard compositions.

The case beginning with 1 is A004760, complement A004754.

The complement is A022340.

These compositions are counted by A099036, complement A212804.

The case covering an initial interval is A333217.

The gapless but non-initial version is A356843, unordered A356845.

Cf. A004780, A005408, A055932, A073492, A073493, A132747.

Sequence in context: A333227 A285510 A174415 * A103826 A361386 A361786

Adjacent sequences: A356841 A356842 A356843 * A356845 A356846 A356847

Keyword

nonn

Author

Gus Wiseman, Sep 02 2022

Status

approved